Theorem resco 5175

Description: Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)

Assertion

Ref	Expression
resco	|- ( ( A o. B ) |` C ) = ( A o. ( B |` C ) )

Proof of Theorem resco

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relres 4975	. 2 |- Rel ( ( A o. B ) |` C )
2		relco 5169	. 2 |- Rel ( A o. ( B |` C ) )
3		vex 2766	. . . . . 6 |- x e. _V
4		vex 2766	. . . . . 6 |- y e. _V
5	3, 4	brco 4838	. . . . 5 |- ( x ( A o. B ) y <-> E. z ( x B z /\ z A y ) )
6	5	anbi1i 458	. . . 4 |- ( ( x ( A o. B ) y /\ x e. C ) <-> ( E. z ( x B z /\ z A y ) /\ x e. C ) )
7		19.41v 1917	. . . 4 |- ( E. z ( ( x B z /\ z A y ) /\ x e. C ) <-> ( E. z ( x B z /\ z A y ) /\ x e. C ) )
8		an32 562	. . . . . 6 |- ( ( ( x B z /\ z A y ) /\ x e. C ) <-> ( ( x B z /\ x e. C ) /\ z A y ) )
9		vex 2766	. . . . . . . 8 |- z e. _V
10	9	brres 4953	. . . . . . 7 |- ( x ( B |` C ) z <-> ( x B z /\ x e. C ) )
11	10	anbi1i 458	. . . . . 6 |- ( ( x ( B |` C ) z /\ z A y ) <-> ( ( x B z /\ x e. C ) /\ z A y ) )
12	8, 11	bitr4i 187	. . . . 5 |- ( ( ( x B z /\ z A y ) /\ x e. C ) <-> ( x ( B |` C ) z /\ z A y ) )
13	12	exbii 1619	. . . 4 |- ( E. z ( ( x B z /\ z A y ) /\ x e. C ) <-> E. z ( x ( B |` C ) z /\ z A y ) )
14	6, 7, 13	3bitr2i 208	. . 3 |- ( ( x ( A o. B ) y /\ x e. C ) <-> E. z ( x ( B |` C ) z /\ z A y ) )
15	4	brres 4953	. . 3 |- ( x ( ( A o. B ) |` C ) y <-> ( x ( A o. B ) y /\ x e. C ) )
16	3, 4	brco 4838	. . 3 |- ( x ( A o. ( B |` C ) ) y <-> E. z ( x ( B |` C ) z /\ z A y ) )
17	14, 15, 16	3bitr4i 212	. 2 |- ( x ( ( A o. B ) |` C ) y <-> x ( A o. ( B |` C ) ) y )
18	1, 2, 17	eqbrriv 4759	1 |- ( ( A o. B ) |` C ) = ( A o. ( B |` C ) )

Syntax hints:   /\ wa 104   = wceq 1364   E.wex 1506   e. wcel 2167   class class class wbr 4034   |` cres 4666   o. ccom 4668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-xp 4670  df-rel 4671  df-co 4673  df-res 4676

This theorem is referenced by:  cocnvcnv2  5182  coires1  5188  relcoi1  5202  dftpos2  6328